Public transportation
If you're staying near the elevators, transportation shouldn't be an issue. In addition, most hotels offer transfers by their own buses or autos. But if you don't want to depend on them, rental cars and cabs are available.
There's also the famous Le Moulet, a free bus that runs through the city center. It runs past the parking lot at the southern entrance to Chamonix, and departs every ten minutes during peak season. There are 19 seats for you, and there are seats for people with disabilities.
There are also buses that visit ski resorts and tourist spots throughout the valley. All lines start from the South Chamonix stop, bypassing the town.
You can travel for free if you have a Carte d'Hote, Gens Du Pays or ski pass. Ask your hotel supplier for a Carte d'Hote card.
Fare
If you don't have such a card, the fare is 1.50 EUR, except for the Pelerins line, which costs only 0.70 EUR per fare.
There is also a night bus for the summer and winter seasons, it does not run between seasons. The trip costs 2 EUR.
In high season, the waiting time is no more than 20 minutes, but it is better to check the schedule.
The website for Chamonix buses is there.
Taxis can be booked there.